 Hi and welcome to the session. I am Shashi and I am going to help you to solve the following question. Question is, hit star b a binary operation on set Q of rational numbers as follows. Third part is a star b is equal to a plus a v. Find which of the binary operations are commutative and which are associative. First of all let us understand the key idea to solve the given question. A binary operation star on the set A is commutative if a star b is equal to b star a for every a b belonging to set A. Now a binary operation star from a cross a to a is said to be associative if a star b star c is equal to a star bracket b star c for every a b c belonging to set A. Let us now start with a solution. We know on Q star is binary operation star is defined as a star b is equal to a plus a b right. First of all let us check if star is commutative. Now we know a star b is equal to a plus a b for every a b belonging to b star a is equal to b plus b a for every a b belonging to Q is not equal to b plus b a for every a b belonging to Q right. And guys a star b is not equal to b star a so we can write binary operation star is not commutative on set Q where Q is the set of all rational numbers. Now let us check if star is associative first of all let us find out a star b star which is equal to a star b plus b c we know b star c is equal to b plus b c so we substitute the value of b star c equal to b plus b c. Now it is further equal to a plus a multiplied by b plus b c we know a star b is equal to a plus a b here a star this whole bracket is our b so we get a plus a b that is a multiplied by this complete bracket so we get a plus a b plus a b c so we can write a star bracket b star c is equal to a plus a b plus a b c. Let us now find out a star a star c it is equal to a plus a b star c now this bracket is our a and this is b so we get a plus a b a plus a b multiplied by c which is further equal to a plus a b plus a c plus a b c so we get a star b star c is equal to a plus a b plus a c plus a b c now we can write plus a b plus a b c is not equal to a plus a b plus a c plus a b c now this is equal to a star bracket b star c this is equal to a star b star c so we get this implies a star bracket b star c is not equal to a star b star c so we can write binary operation star is not associative on q where q is the set of all rational numbers so our final answer is binary operation star is neither commutative nor associative this is our final answer hope you understood the session take care and goodbye.